From texture painting to UV map
I will start the example with creating a texture from a box mapped image texture set to blend as a way to get something baked to the initial uv map.

I've set up an image texture to bake to as a target and am using an emission shader to avoid shadow

I leave the image target disconnected and selected before baking Emit

Here you see in the UV editor the newly baked pixels on the target image in relation to the UV map, save this image now to disk

Here we set up a New UV map and set it to viewable, then unwrap to a new scheme

Here we check the new mapping to a new image texture target so you can see we are not dealing with the previous UV map - reference the new map in the UV node

We need to reverse the connection so that we leave the target image node disconnected and still selected and then press bake emit

Our result on the new UV map is visible now, and we have gone from generated texture coordinate image texture to UV mapped image texture to a second UV mapped image texture. Save you result image to disk if satisfied with the mapping.